Compatibility of Kazhdan and Brauer homomorphism

Abstract

Let $G$ be a connected split reductive group defined over $\mathbb{Z}$. Let $F$ and $F^{\prime }$ be two non-Archimedean $m$-close local fields, where $m$ is a positive integer. D.Kazhdan gave an isomorphism between the Hecke algebras ${\mathrm{Kaz}}_m^F:\mathcal{H}(G(F),K_F)\to \mathcal{H}(G(F^{\prime }),K_{F^{\prime }})$, where $K_F$ and $K_{F^{\prime }}$ are the $m$-th usual congruence subgroups of $G(F)$ and $G(F^{\prime })$ respectively. On the other hand, if $\sigma$ is an automorphism of $G$ of prime order $l$, then we have Brauer homomorphism $\mathrm{Br}:\mathcal{H}(G(F),U(F))\to \mathcal{H}(G^{\sigma }(F),U^{\sigma }(F))$, where $U(F)$ and $U^{\sigma }(F)$ are compact open subgroups of $G(F)$ and $G^{\sigma }(F)$ respectively. In this article, we study the compatibility between these two maps in the local base change setting. Further, an application of this compatibility is given in the context of linkage--which is the representation theoretic version of Brauer homomorphism.

14 pages, final version. To appear in Canadian Math. Bulletin

References in corpus (1)

• A Hecke algebra isomorphism over close local fields